Index

A (appendix), 1
A (archetype), 2
A (definition), 3
A (notation), 4
A (part), 5
AA (Property), 6
AA (subsection, section WILA), 7
AA (theorem), 8
AAC (Property), 9
AACN (Property), 10
AAF (Property), 11
AALC (example), 12
AAM (Property), 13
ABLC (example), 14
ABS (example), 15
AC (Property), 16
ACC (Property), 17
ACCN (Property), 18
ACF (Property), 19
ACM (Property), 20
ACN (example), 21
additive associativity
    column vectors
        Property AAC, 22
    complex numbers
        Property AACN, 23
    matrices
        Property AAM, 24
    vectors
        Property AA, 25
additive closure
    column vectors
        Property ACC, 26
    complex numbers
        Property ACCN, 27
    field
        Property ACF, 28
    matrices
        Property ACM, 29
    vectors
        Property AC, 30
additive commutativity
    complex numbers
        Property CACN, 31
additive inverse
    complex numbers
        Property AICN, 32
    from scalar multiplication
        theorem AISM, 33
additive inverses
    column vectors
        Property AIC, 34
    matrices
        Property AIM, 35
    unique
        theorem AIU, 36
    vectors
        Property AI, 37
adjoint
    definition A, 38
    inner product
        theorem AIP, 39
    notation, 40
    of a matrix sum
        theorem AMA, 41
    of an adjoint
        theorem AA, 42
    of matrix scalar multiplication
        theorem AMSM, 43
AHSAC (example), 44
AI (Property), 45
AIC (Property), 46
AICN (Property), 47
AIF (Property), 48
AIM (Property), 49
AIP (theorem), 50
AISM (theorem), 51
AIU (theorem), 52
AIVLT (example), 53
ALT (example), 54
ALTMM (example), 55
AM (definition), 56
AM (example), 57
AM (notation), 58
AM (subsection, section MO), 59
AMA (theorem), 60
AMAA (example), 61
AME (definition), 62
AME (notation), 63
AMSM (theorem), 64
ANILT (example), 65
ANM (example), 66
AOS (example), 67
Archetype A
    column space, 68
    linearly dependent columns, 69
    singular matrix, 70
    solving homogeneous system, 71
    system as linear combination, 72
archetype A
    augmented matrix
        example AMAA, 73
Archetype B
    column space, 74
    inverse
        example CMIAB, 75
    linearly independent columns, 76
    nonsingular matrix, 77
    not invertible
        example MWIAA, 78
    solutions via inverse
        example SABMI, 79
    solving homogeneous system, 80
    system as linear combination, 81
    vector equality, 82
archetype B
    solutions
        example SAB, 83
Archetype C
    homogeneous system, 84
Archetype D
    column space, original columns, 85
    solving homogeneous system, 86
    vector form of solutions, 87
Archetype I
    column space from row operations, 88
    null space, 89
    row space, 90
    vector form of solutions, 91
Archetype I:casting out vectors, 92
Archetype L
    null space span, linearly independent, 93
    vector form of solutions, 94
ASC (example), 95
augmented matrix
    notation, 96
AVR (example), 97

B (archetype), 98
B (definition), 99
B (section), 100
B (subsection, section B), 101
basis
    columns nonsingular matrix
        example CABAK, 102
    common size
        theorem BIS, 103
    crazy vector apace
        example BC, 104
    definition B, 105
    matrices
        example BM, 106
        example BSM22, 107
    polynomials
        example BP, 108
        example BPR, 109
        example BSP4, 110
        example SVP4, 111
    subspace of matrices
        example BDM22, 112
BC (example), 113
BCS (theorem), 114
BDE (example), 115
BDM22 (example), 116
best cities
    money magazine
        example MBC, 117
BIS (theorem), 118
BM (example), 119
BNM (subsection, section B), 120
BNS (theorem), 121
BP (example), 122
BPR (example), 123
BRLT (example), 124
BRS (theorem), 125
BS (theorem), 126
BSCV (subsection, section B), 127
BSM22 (example), 128
BSP4 (example), 129

C (archetype), 130
C (definition), 131
C (notation), 132
C (part), 133
C (Property), 134
C (technique, section PT), 135
CABAK (example), 136
CACN (Property), 137
CAEHW (example), 138
CAF (Property), 139
canonical form
    nilpotent linear transformation
        example CFNLT, 140
        theorem CFNLT, 141
CAV (subsection, section O), 142
Cayley-Hamilton
    theorem CHT, 143
CB (section), 144
CB (theorem), 145
CBCV (example), 146
CBM (definition), 147
CBM (subsection, section CB), 148
CBP (example), 149
CC (Property), 150
CCCV (definition), 151
CCCV (notation), 152
CCM (definition), 153
CCM (example), 154
CCM (notation), 155
CCM (theorem), 156
CCN (definition), 157
CCN (notation), 158
CCN (subsection, section CNO), 159
CCRA (theorem), 160
CCRM (theorem), 161
CCT (theorem), 162
CD (subsection, section DM), 163
CD (technique, section PT), 164
CEE (subsection, section EE), 165
CELT (example), 166
CELT (subsection, section CB), 167
CEMS6 (example), 168
CF (section), 169
CFDVS (theorem), 170
CFNLT (example), 171
CFNLT (subsection, section NLT), 172
CFNLT (theorem), 173
CFV (example), 174
change of basis
    between polynomials
        example CBP, 175
change-of-basis
    between column vectors
        example CBCV, 176
    matrix representation
        theorem MRCB, 177
    similarity
        theorem SCB, 178
    theorem CB, 179
change-of-basis matrix
    definition CBM, 180
    inverse
        theorem ICBM, 181
characteristic polynomial
    definition CP, 182
    degree
        theorem DCP, 183
    size 3 matrix
        example CPMS3, 184
CHT (subsection, section JCF), 185
CHT (theorem), 186
CILT (subsection, section ILT), 187
CILTI (theorem), 188
CIM (subsection, section MISLE), 189
CINM (theorem), 190
CIVLT (example), 191
CIVLT (theorem), 192
CLI (theorem), 193
CLTLT (theorem), 194
CM (definition), 195
CM (Property), 196
CM32 (example), 197
CMCN (Property), 198
CMF (Property), 199
CMI (example), 200
CMIAB (example), 201
CMVEI (theorem), 202
CN (appendix), 203
CNA (definition), 204
CNA (notation), 205
CNA (subsection, section CNO), 206
CNE (definition), 207
CNE (notation), 208
CNM (definition), 209
CNM (notation), 210
CNMB (theorem), 211
CNO (section), 212
CNS1 (example), 213
CNS2 (example), 214
CNSV (example), 215
COB (theorem), 216
coefficient matrix
    definition CM, 217
    nonsingular
        theorem SNCM, 218
column space
    as null space
        theorem FS, 219
    Archetype A
        example CSAA, 220
    Archetype B
        example CSAB, 221
    as null space
        example CSANS, 222
    as null space, Archetype G
        example FSAG, 223
    as row space
        theorem CSRST, 224
    basis
        theorem BCS, 225
    consistent system
        theorem CSCS, 226
    consistent systems
        example CSMCS, 227
    isomorphic to range, 228
    matrix, 229
    nonsingular matrix
        theorem CSNM, 230
    notation, 231
    original columns, Archetype D
        example CSOCD, 232
    row operations, Archetype I
        example CSROI, 233
    subspace
        theorem CSMS, 234
    testing membership
        example MCSM, 235
    two computations
        example CSTW, 236
column vector addition
    notation, 237
column vector scalar multiplication
    notation, 238
commutativity
    column vectors
        Property CC, 239
    matrices
        Property CM, 240
    vectors
        Property C, 241
complex m-space
    example VSCV, 242
complex arithmetic
    example ACN, 243
complex number
    conjugate
        example CSCN, 244
    modulus
        example MSCN, 245
complex number
    conjugate
        definition CCN, 246
    modulus
        definition MCN, 247
complex numbers
    addition
        definition CNA, 248
        notation, 249
    arithmetic properties
        theorem PCNA, 250
    equality
        definition CNE, 251
        notation, 252
    multiplication
        definition CNM, 253
        notation, 254
complex vector space
    dimension
        theorem DCM, 255
composition
    injective linear transformations
        theorem CILTI, 256
    surjective linear transformations
        theorem CSLTS, 257
conjugate
    addition
        theorem CCRA, 258
    column vector
        definition CCCV, 259
    matrix
        definition CCM, 260
        notation, 261
    multiplication
        theorem CCRM, 262
    notation, 263
    of conjugate of a matrix
        theorem CCM, 264
    scalar multiplication
        theorem CRSM, 265
    twice
        theorem CCT, 266
    vector addition
        theorem CRVA, 267
conjugate of a vector
    notation, 268
conjugation
    matrix addition
        theorem CRMA, 269
    matrix scalar multiplication
        theorem CRMSM, 270
    matrix transpose
        theorem MCT, 271
consistent linear system, 272
consistent linear systems
    theorem CSRN, 273
consistent system
    definition CS, 274
constructive proofs
    technique C, 275
contradiction
    technique CD, 276
contrapositive
    technique CP, 277
converse
    technique CV, 278
coordinates
    orthonormal basis
        theorem COB, 279
coordinatization
    linear combination of matrices
        example CM32, 280
    linear independence
        theorem CLI, 281
    orthonormal basis
        example CROB3, 282
        example CROB4, 283
    spanning sets
        theorem CSS, 284
coordinatization principle, 285
coordinatizing
    polynomials
        example CP2, 286
COV (example), 287
COV (subsection, section LDS), 288
CP (definition), 289
CP (subsection, section VR), 290
CP (technique, section PT), 291
CP2 (example), 292
CPMS3 (example), 293
CPSM (theorem), 294
crazy vector space
    example CVSR, 295
    properties
        example PCVS, 296
CRMA (theorem), 297
CRMSM (theorem), 298
CRN (theorem), 299
CROB3 (example), 300
CROB4 (example), 301
CRS (section), 302
CRS (subsection, section FS), 303
CRSM (theorem), 304
CRVA (theorem), 305
CS (definition), 306
CS (example), 307
CS (subsection, section TSS), 308
CSAA (example), 309
CSAB (example), 310
CSANS (example), 311
CSCN (example), 312
CSCS (theorem), 313
CSIP (example), 314
CSLT (subsection, section SLT), 315
CSLTS (theorem), 316
CSM (definition), 317
CSM (notation), 318
CSMCS (example), 319
CSMS (theorem), 320
CSNM (subsection, section CRS), 321
CSNM (theorem), 322
CSOCD (example), 323
CSRN (theorem), 324
CSROI (example), 325
CSRST (diagram), 326
CSRST (theorem), 327
CSS (theorem), 328
CSSE (subsection, section CRS), 329
CSSOC (subsection, section CRS), 330
CSTW (example), 331
CTD (subsection, section TD), 332
CTLT (example), 333
CUMOS (theorem), 334
curve fitting
    polynomial through 5 points
        example PTFP, 335
CV (definition), 336
CV (notation), 337
CV (technique, section PT), 338
CVA (definition), 339
CVA (notation), 340
CVC (notation), 341
CVE (definition), 342
CVE (notation), 343
CVS (example), 344
CVS (subsection, section VR), 345
CVSM (definition), 346
CVSM (example), 347
CVSM (notation), 348
CVSR (example), 349

D (acronyms, section PDM), 350
D (archetype), 351
D (chapter), 352
D (definition), 353
D (notation), 354
D (section), 355
D (subsection, section D), 356
D (subsection, section SD), 357
D (technique, section PT), 358
D33M (example), 359
DAB (example), 360
DC (example), 361
DC (technique, section PT), 362
DC (theorem), 363
DCM (theorem), 364
DCN (Property), 365
DCP (theorem), 366
DD (subsection, section DM), 367
DEC (theorem), 368
decomposition
    technique DC, 369
DED (theorem), 370
definition
    A, 371
    AM, 372
    AME, 373
    B, 374
    C, 375
    CBM, 376
    CCCV, 377
    CCM, 378
    CCN, 379
    CM, 380
    CNA, 381
    CNE, 382
    CNM, 383
    CP, 384
    CS, 385
    CSM, 386
    CV, 387
    CVA, 388
    CVE, 389
    CVSM, 390
    D, 391
    DIM, 392
    DM, 393
    DS, 394
    DZM, 395
    EEF, 396
    EELT, 397
    EEM, 398
    ELEM, 399
    EM, 400
    EO, 401
    ES, 402
    ESYS, 403
    F, 404
    GES, 405
    GEV, 406
    GME, 407
    HI, 408
    HID, 409
    HM, 410
    HP, 411
    HS, 412
    IDLT, 413
    IDV, 414
    IE, 415
    ILT, 416
    IM, 417
    IMP, 418
    IP, 419
    IS, 420
    IVLT, 421
    IVS, 422
    JB, 423
    JCF, 424
    KLT, 425
    LC, 426
    LCCV, 427
    LI, 428
    LICV, 429
    LNS, 430
    LSS, 431
    LT, 432
    LTA, 433
    LTC, 434
    LTM, 435
    LTR, 436
    LTSM, 437
    M, 438
    MA, 439
    MCN, 440
    ME, 441
    MI, 442
    MM, 443
    MR, 444
    MRLS, 445
    MSM, 446
    MVP, 447
    NLT, 448
    NM, 449
    NOLT, 450
    NOM, 451
    NRML, 452
    NSM, 453
    NV, 454
    ONS, 455
    OSV, 456
    OV, 457
    PI, 458
    PSM, 459
    REM, 460
    RLD, 461
    RLDCV, 462
    RLT, 463
    RO, 464
    ROLT, 465
    ROM, 466
    RR, 467
    RREF, 468
    RSM, 469
    S, 470
    SC, 471
    SE, 472
    SET, 473
    SI, 474
    SIM, 475
    SLE, 476
    SLT, 477
    SM, 478
    SOLV, 479
    SQM, 480
    SRM, 481
    SS, 482
    SSCV, 483
    SSET, 484
    SSLE, 485
    SSSLE, 486
    SU, 487
    SUV, 488
    SV, 489
    SYM, 490
    T, 491
    technique D, 492
    TM, 493
    TS, 494
    TSHSE, 495
    TSVS, 496
    UM, 497
    UTM, 498
    VM, 499
    VOC, 500
    VR, 501
    VS, 502
    VSCV, 503
    VSM, 504
    ZCV, 505
    ZM, 506
DEHD (example), 507
DEM (theorem), 508
DEMMM (theorem), 509
DEMS5 (example), 510
DER (theorem), 511
DERC (theorem), 512
determinant
    computed two ways
        example TCSD, 513
    definition DM, 514
    equal rows or columns
        theorem DERC, 515
    expansion, columns
        theorem DEC, 516
    expansion, rows
        theorem DER, 517
    identity matrix
        theorem DIM, 518
    matrix multiplication
        theorem DRMM, 519
    nonsingular matrix, 520
    notation, 521
    row or column multiple
        theorem DRCM, 522
    row or column swap
        theorem DRCS, 523
    size 2 matrix
        theorem DMST, 524
    size 3 matrix
        example D33M, 525
    transpose
        theorem DT, 526
    via row operations
        example DRO, 527
    zero
        theorem SMZD, 528
    zero row or column
        theorem DZRC, 529
    zero versus nonzero
        example ZNDAB, 530
determinant, upper triangular matrix
    example DUTM, 531
determinants
    elementary matrices
        theorem DEMMM, 532
DF (Property), 533
DF (subsection, section CF), 534
DFS (subsection, section PD), 535
DFS (theorem), 536
DGES (theorem), 537
diagonal matrix
    definition DIM, 538
diagonalizable
    definition DZM, 539
    distinct eigenvalues
        example DEHD, 540
        theorem DED, 541
    full eigenspaces
        theorem DMFE, 542
    not
        example NDMS4, 543
diagonalizable matrix
    high power
        example HPDM, 544
diagonalization
    Archetype B
        example DAB, 545
    criteria
        theorem DC, 546
    example DMS3, 547
diagram
    CSRST, 548
    DLTA, 549
    DLTM, 550
    DTSLS, 551
    FTMR, 552
    FTMRA, 553
    GLT, 554
    ILT, 555
    MRCLT, 556
    NILT, 557
DIM (definition), 558
DIM (theorem), 559
dimension
    crazy vector space
        example DC, 560
    definition D, 561
    notation, 562
    polynomial subspace
        example DSP4, 563
    proper subspaces
        theorem PSSD, 564
    subspace
        example DSM22, 565
direct sum
    decomposing zero vector
        theorem DSZV, 566
    definition DS, 567
    dimension
        theorem DSD, 568
    example SDS, 569
    from a basis
        theorem DSFB, 570
    from one subspace
        theorem DSFOS, 571
    notation, 572
    zero intersection
        theorem DSZI, 573
direct sums
    linear independence
        theorem DSLI, 574
    repeated
        theorem RDS, 575
distributivity
    complex numbers
        Property DCN, 576
    field
        Property DF, 577
distributivity, matrix addition
    matrices
        Property DMAM, 578
distributivity, scalar addition
    column vectors
        Property DSAC, 579
    matrices
        Property DSAM, 580
    vectors
        Property DSA, 581
distributivity, vector addition
    column vectors
        Property DVAC, 582
    vectors
        Property DVA, 583
DLDS (theorem), 584
DLTA (diagram), 585
DLTM (diagram), 586
DM (definition), 587
DM (notation), 588
DM (section), 589
DM (theorem), 590
DMAM (Property), 591
DMFE (theorem), 592
DMHP (subsection, section HP), 593
DMHP (theorem), 594
DMMP (theorem), 595
DMS3 (example), 596
DMST (theorem), 597
DNLT (theorem), 598
DNMMM (subsection, section PDM), 599
DP (theorem), 600
DRCM (theorem), 601
DRCMA (theorem), 602
DRCS (theorem), 603
DRMM (theorem), 604
DRO (example), 605
DRO (subsection, section PDM), 606
DROEM (subsection, section PDM), 607
DS (definition), 608
DS (notation), 609
DS (subsection, section PD), 610
DSA (Property), 611
DSAC (Property), 612
DSAM (Property), 613
DSD (theorem), 614
DSFB (theorem), 615
DSFOS (theorem), 616
DSLI (theorem), 617
DSM22 (example), 618
DSP4 (example), 619
DSZI (theorem), 620
DSZV (theorem), 621
DT (theorem), 622
DTSLS (diagram), 623
DUTM (example), 624
DVA (Property), 625
DVAC (Property), 626
DVM (theorem), 627
DVS (subsection, section D), 628
DZM (definition), 629
DZRC (theorem), 630

E (acronyms, section SD), 631
E (archetype), 632
E (chapter), 633
E (technique, section PT), 634
E.SAGE (computation, section SAGE), 635
ECEE (subsection, section EE), 636
EDELI (theorem), 637
EDYES (theorem), 638
EE (section), 639
EEE (subsection, section EE), 640
EEF (definition), 641
EEF (subsection, section FS), 642
EELT (definition), 643
EELT (subsection, section CB), 644
EEM (definition), 645
EEM (subsection, section EE), 646
EEMAP (theorem), 647
EENS (example), 648
EER (theorem), 649
EESR (theorem), 650
EHM (subsection, section PEE), 651
eigenspace
    as null space
        theorem EMNS, 652
    definition EM, 653
    invariant subspace
        theorem EIS, 654
    subspace
        theorem EMS, 655
eigenspaces
    sage, 656
eigenvalue
    algebraic multiplicity
        definition AME, 657
        notation, 658
    complex
        example CEMS6, 659
    definition EEM, 660
    existence
        example CAEHW, 661
        theorem EMHE, 662
    geometric multiplicity
        definition GME, 663
        notation, 664
    index, 665
    linear transformation
        definition EELT, 666
    multiplicities
        example EMMS4, 667
    power
        theorem EOMP, 668
    root of characteristic polynomial
        theorem EMRCP, 669
    scalar multiple
        theorem ESMM, 670
    symmetric matrix
        example ESMS4, 671
    zero
        theorem SMZE, 672
eigenvalues
    building desired
        example BDE, 673
    complex, of a linear transformation
        example CELT, 674
    conjugate pairs
        theorem ERMCP, 675
    distinct
        example DEMS5, 676
    example SEE, 677
    Hermitian matrices
        theorem HMRE, 678
    inverse
        theorem EIM, 679
    maximum number
        theorem MNEM, 680
    multiplicities
        example HMEM5, 681
        theorem ME, 682
    number
        theorem NEM, 683
    of a polynomial
        theorem EPM, 684
    size 3 matrix
        example EMS3, 685
        example ESMS3, 686
    transpose
        theorem ETM, 687
eigenvalues, eigenvectors
    vector, matrix representations
        theorem EER, 688
eigenvector, 689
    linear transformation, 690
eigenvectors, 691
    conjugate pairs, 692
    Hermitian matrices
        theorem HMOE, 693
    linear transformation
        example ELTBM, 694
        example ELTBP, 695
    linearly independent
        theorem EDELI, 696
    of a linear transformation
        example ELTT, 697
EILT (subsection, section ILT), 698
EIM (theorem), 699
EIS (example), 700
EIS (theorem), 701
ELEM (definition), 702
ELEM (notation), 703
elementary matrices
    definition ELEM, 704
    determinants
        theorem DEM, 705
    nonsingular
        theorem EMN, 706
    notation, 707
    row operations
        example EMRO, 708
        theorem EMDRO, 709
ELIS (theorem), 710
ELTBM (example), 711
ELTBP (example), 712
ELTT (example), 713
EM (definition), 714
EM (subsection, section DM), 715
EMDRO (theorem), 716
EMHE (theorem), 717
EMMS4 (example), 718
EMMVP (theorem), 719
EMN (theorem), 720
EMNS (theorem), 721
EMP (theorem), 722
empty set, 723
    notation, 724
EMRCP (theorem), 725
EMRO (example), 726
EMS (theorem), 727
EMS3 (example), 728
ENLT (theorem), 729
EO (definition), 730
EOMP (theorem), 731
EOPSS (theorem), 732
EPM (theorem), 733
EPSM (theorem), 734
equal matrices
    via equal matrix-vector products
        theorem EMMVP, 735
equation operations
    definition EO, 736
    theorem EOPSS, 737
equivalence statements
    technique E, 738
equivalences
    technique ME, 739
equivalent systems
    definition ESYS, 740
ERMCP (theorem), 741
ES (definition), 742
ES (notation), 743
ESEO (subsection, section SSLE), 744
ESLT (subsection, section SLT), 745
ESMM (theorem), 746
ESMS3 (example), 747
ESMS4 (example), 748
ESYS (definition), 749
ETM (theorem), 750
EVS (subsection, section VS), 751
example
    AALC, 752
    ABLC, 753
    ABS, 754
    ACN, 755
    AHSAC, 756
    AIVLT, 757
    ALT, 758
    ALTMM, 759
    AM, 760
    AMAA, 761
    ANILT, 762
    ANM, 763
    AOS, 764
    ASC, 765
    AVR, 766
    BC, 767
    BDE, 768
    BDM22, 769
    BM, 770
    BP, 771
    BPR, 772
    BRLT, 773
    BSM22, 774
    BSP4, 775
    CABAK, 776
    CAEHW, 777
    CBCV, 778
    CBP, 779
    CCM, 780
    CELT, 781
    CEMS6, 782
    CFNLT, 783
    CFV, 784
    CIVLT, 785
    CM32, 786
    CMI, 787
    CMIAB, 788
    CNS1, 789
    CNS2, 790
    CNSV, 791
    COV, 792
    CP2, 793
    CPMS3, 794
    CROB3, 795
    CROB4, 796
    CS, 797
    CSAA, 798
    CSAB, 799
    CSANS, 800
    CSCN, 801
    CSIP, 802
    CSMCS, 803
    CSOCD, 804
    CSROI, 805
    CSTW, 806
    CTLT, 807
    CVS, 808
    CVSM, 809
    CVSR, 810
    D33M, 811
    DAB, 812
    DC, 813
    DEHD, 814
    DEMS5, 815
    DMS3, 816
    DRO, 817
    DSM22, 818
    DSP4, 819
    DUTM, 820
    EENS, 821
    EIS, 822
    ELTBM, 823
    ELTBP, 824
    ELTT, 825
    EMMS4, 826
    EMRO, 827
    EMS3, 828
    ESMS3, 829
    ESMS4, 830
    FDV, 831
    FF8, 832
    FRAN, 833
    FS1, 834
    FS2, 835
    FSAG, 836
    FSCF, 837
    GE4, 838
    GE6, 839
    GENR6, 840
    GSTV, 841
    HISAA, 842
    HISAD, 843
    HMEM5, 844
    HP, 845
    HPDM, 846
    HUSAB, 847
    IAP, 848
    IAR, 849
    IAS, 850
    IAV, 851
    ILTVR, 852
    IM, 853
    IM11, 854
    IS, 855
    ISJB, 856
    ISMR4, 857
    ISMR6, 858
    ISSI, 859
    IVSAV, 860
    JB4, 861
    JCF10, 862
    KPNLT, 863
    KVMR, 864
    LCM, 865
    LDCAA, 866
    LDHS, 867
    LDP4, 868
    LDRN, 869
    LDS, 870
    LIC, 871
    LICAB, 872
    LIHS, 873
    LIM32, 874
    LINSB, 875
    LIP4, 876
    LIS, 877
    LLDS, 878
    LNS, 879
    LTDB1, 880
    LTDB2, 881
    LTDB3, 882
    LTM, 883
    LTPM, 884
    LTPP, 885
    LTRGE, 886
    MA, 887
    MBC, 888
    MCSM, 889
    MFLT, 890
    MI, 891
    MIVS, 892
    MMNC, 893
    MNSLE, 894
    MOLT, 895
    MPMR, 896
    MRBE, 897
    MRCM, 898
    MSCN, 899
    MSM, 900
    MTV, 901
    MWIAA, 902
    NDMS4, 903
    NIAO, 904
    NIAQ, 905
    NIAQR, 906
    NIDAU, 907
    NJB5, 908
    NKAO, 909
    NLT, 910
    NM, 911
    NM62, 912
    NM64, 913
    NM83, 914
    NRREF, 915
    NSAO, 916
    NSAQ, 917
    NSAQR, 918
    NSC2A, 919
    NSC2S, 920
    NSC2Z, 921
    NSDAT, 922
    NSDS, 923
    NSE, 924
    NSEAI, 925
    NSLE, 926
    NSLIL, 927
    NSNM, 928
    NSR, 929
    NSS, 930
    OLTTR, 931
    ONFV, 932
    ONTV, 933
    OSGMD, 934
    OSMC, 935
    PCVS, 936
    PM, 937
    PSHS, 938
    PTFP, 939
    PTM, 940
    PTMEE, 941
    RAO, 942
    RES, 943
    RNM, 944
    RNSM, 945
    ROD2, 946
    ROD4, 947
    RREF, 948
    RREFN, 949
    RRTI, 950
    RS, 951
    RSAI, 952
    RSB, 953
    RSC4, 954
    RSC5, 955
    RSNS, 956
    RSREM, 957
    RVMR, 958
    S, 959
    SAA, 960
    SAB, 961
    SABMI, 962
    SAE, 963
    SAN, 964
    SAR, 965
    SAV, 966
    SC, 967
    SC3, 968
    SCAA, 969
    SCAB, 970
    SCAD, 971
    SDS, 972
    SEE, 973
    SEEF, 974
    SETM, 975
    SI, 976
    SM2Z7, 977
    SM32, 978
    SMLT, 979
    SMS3, 980
    SMS5, 981
    SP4, 982
    SPIAS, 983
    SRR, 984
    SS, 985
    SS6W, 986
    SSC, 987
    SSET, 988
    SSM22, 989
    SSNS, 990
    SSP, 991
    SSP4, 992
    STLT, 993
    STNE, 994
    SU, 995
    SUVOS, 996
    SVP4, 997
    SYM, 998
    TCSD, 999
    TD4, 1000
    TDEE6, 1001
    TDSSE, 1002
    TIS, 1003
    TIVS, 1004
    TKAP, 1005
    TLC, 1006
    TM, 1007
    TMP, 1008
    TOV, 1009
    TREM, 1010
    TTS, 1011
    UM3, 1012
    UPM, 1013
    US, 1014
    USR, 1015
    VA, 1016
    VESE, 1017
    VFS, 1018
    VFSAD, 1019
    VFSAI, 1020
    VFSAL, 1021
    VM4, 1022
    VRC4, 1023
    VRP2, 1024
    VSCV, 1025
    VSF, 1026
    VSIM5, 1027
    VSIS, 1028
    VSM, 1029
    VSP, 1030
    VSPUD, 1031
    VSS, 1032
    ZNDAB, 1033
EXC (subsection, section B), 1034
EXC (subsection, section CB), 1035
EXC (subsection, section CF), 1036
EXC (subsection, section CRS), 1037
EXC (subsection, section D), 1038
EXC (subsection, section DM), 1039
EXC (subsection, section EE), 1040
EXC (subsection, section F), 1041
EXC (subsection, section FS), 1042
EXC (subsection, section HP), 1043
EXC (subsection, section HSE), 1044
EXC (subsection, section ILT), 1045
EXC (subsection, section IS), 1046
EXC (subsection, section IVLT), 1047
EXC (subsection, section LC), 1048
EXC (subsection, section LDS), 1049
EXC (subsection, section LI), 1050
EXC (subsection, section LISS), 1051
EXC (subsection, section LT), 1052
EXC (subsection, section MINM), 1053
EXC (subsection, section MISLE), 1054
EXC (subsection, section MM), 1055
EXC (subsection, section MO), 1056
EXC (subsection, section MR), 1057
EXC (subsection, section NM), 1058
EXC (subsection, section O), 1059
EXC (subsection, section PD), 1060
EXC (subsection, section PDM), 1061
EXC (subsection, section PEE), 1062
EXC (subsection, section PSM), 1063
EXC (subsection, section RREF), 1064
EXC (subsection, section S), 1065
EXC (subsection, section SD), 1066
EXC (subsection, section SLT), 1067
EXC (subsection, section SS), 1068
EXC (subsection, section SSLE), 1069
EXC (subsection, section T), 1070
EXC (subsection, section TSS), 1071
EXC (subsection, section VO), 1072
EXC (subsection, section VR), 1073
EXC (subsection, section VS), 1074
EXC (subsection, section WILA), 1075
extended echelon form
    submatrices
        example SEEF, 1076
extended reduced row-echelon form
    properties
        theorem PEEF, 1077

F (archetype), 1078
F (definition), 1079
F (section), 1080
F (subsection, section F), 1081
FDV (example), 1082
FF (subsection, section F), 1083
FF8 (example), 1084
Fibonacci sequence
    example FSCF, 1085
field
    definition F, 1086
FIMP (theorem), 1087
finite field
    size 8
        example FF8, 1088
four subsets
    example FS1, 1089
    example FS2, 1090
four subspaces
    dimension
        theorem DFS, 1091
FRAN (example), 1092
free variables
    example CFV, 1093
free variables, number
    theorem FVCS, 1094
free, independent variables
    example FDV, 1095
FS (section), 1096
FS (subsection, section FS), 1097
FS (subsection, section SD), 1098
FS (theorem), 1099
FS1 (example), 1100
FS2 (example), 1101
FSAG (example), 1102
FSCF (example), 1103
FTMR (diagram), 1104
FTMR (theorem), 1105
FTMRA (diagram), 1106
FV (subsection, section TSS), 1107
FVCS (theorem), 1108

G (archetype), 1109
G (theorem), 1110
GE4 (example), 1111
GE6 (example), 1112
GEE (subsection, section IS), 1113
GEK (theorem), 1114
generalized eigenspace
    as kernel
        theorem GEK, 1115
    definition GES, 1116
    dimension
        theorem DGES, 1117
    dimension 4 domain
        example GE4, 1118
    dimension 6 domain
        example GE6, 1119
    invariant subspace
        theorem GESIS, 1120
    nilpotent restriction
        theorem RGEN, 1121
    nilpotent restrictions, dimension 6 domain
        example GENR6, 1122
    notation, 1123
generalized eigenspace decomposition
    theorem GESD, 1124
generalized eigenvector
    definition GEV, 1125
GENR6 (example), 1126
GES (definition), 1127
GES (notation), 1128
GESD (subsection, section JCF), 1129
GESD (theorem), 1130
GESIS (theorem), 1131
GEV (definition), 1132
GFDL (appendix), 1133
GLT (diagram), 1134
GME (definition), 1135
GME (notation), 1136
goldilocks
    theorem G, 1137
Gram-Schmidt
    column vectors
        theorem GSP, 1138
    three vectors
        example GSTV, 1139
gram-schmidt
    mathematica, 1140
GS (technique, section PT), 1141
GSP (subsection, section O), 1142
GSP (theorem), 1143
GSP.MMA (computation, section MMA), 1144
GSTV (example), 1145
GT (subsection, section PD), 1146

H (archetype), 1147
Hadamard Identity
    notation, 1148
Hadamard identity
    definition HID, 1149
Hadamard Inverse
    notation, 1150
Hadamard inverse
    definition HI, 1151
Hadamard Product
    Diagonalizable Matrices
        theorem DMHP, 1152
    notation, 1153
Hadamard product
    commutativity
        theorem HPC, 1154
    definition HP, 1155
    diagonal matrices
        theorem DMMP, 1156
    distributivity
        theorem HPDAA, 1157
    example HP, 1158
    identity
        theorem HPHID, 1159
    inverse
        theorem HPHI, 1160
    scalar matrix multiplication
        theorem HPSMM, 1161
hermitian
    definition HM, 1162
Hermitian matrix
    inner product
        theorem HMIP, 1163
HI (definition), 1164
HI (notation), 1165
HID (definition), 1166
HID (notation), 1167
HISAA (example), 1168
HISAD (example), 1169
HM (definition), 1170
HM (subsection, section MM), 1171
HMEM5 (example), 1172
HMIP (theorem), 1173
HMOE (theorem), 1174
HMRE (theorem), 1175
HMVEI (theorem), 1176
homogeneous system
    Archetype C
        example AHSAC, 1177
    consistent
        theorem HSC, 1178
    definition HS, 1179
    infinitely many solutions
        theorem HMVEI, 1180
homogeneous systems
    linear independence, 1181
HP (definition), 1182
HP (example), 1183
HP (notation), 1184
HP (section), 1185
HPC (theorem), 1186
HPDAA (theorem), 1187
HPDM (example), 1188
HPHI (theorem), 1189
HPHID (theorem), 1190
HPSMM (theorem), 1191
HS (definition), 1192
HSC (theorem), 1193
HSE (section), 1194
HUSAB (example), 1195

I (archetype), 1196
I (technique, section PT), 1197
IAP (example), 1198
IAR (example), 1199
IAS (example), 1200
IAV (example), 1201
ICBM (theorem), 1202
ICLT (theorem), 1203
identities
    technique PI, 1204
identity matrix
    determinant, 1205
    example IM, 1206
    notation, 1207
IDLT (definition), 1208
IDV (definition), 1209
IE (definition), 1210
IE (notation), 1211
IFDVS (theorem), 1212
IILT (theorem), 1213
ILT (definition), 1214
ILT (diagram), 1215
ILT (section), 1216
ILTB (theorem), 1217
ILTD (subsection, section ILT), 1218
ILTD (theorem), 1219
ILTIS (theorem), 1220
ILTLI (subsection, section ILT), 1221
ILTLI (theorem), 1222
ILTLT (theorem), 1223
ILTVR (example), 1224
IM (definition), 1225
IM (example), 1226
IM (notation), 1227
IM (subsection, section MISLE), 1228
IM11 (example), 1229
IMILT (theorem), 1230
IMP (definition), 1231
IMR (theorem), 1232
inconsistent linear systems
    theorem ISRN, 1233
independent, dependent variables
    definition IDV, 1234
indesxstring
    example SM2Z7, 1235
    example SSET, 1236
index
    eigenvalue
        definition IE, 1237
        notation, 1238
indexstring
    theorem DRCMA, 1239
    theorem OBUTR, 1240
    theorem UMCOB, 1241
induction
    technique I, 1242
infinite solution set
    example ISSI, 1243
infinite solutions, 3 × 4
    example IS, 1244
injective
    example IAP, 1245
    example IAR, 1246
    not
        example NIAO, 1247
        example NIAQ, 1248
        example NIAQR, 1249
    not, by dimension
        example NIDAU, 1250
    polynomials to matrices
        example IAV, 1251
injective linear transformation
    bases
        theorem ILTB, 1252
injective linear transformations
    dimension
        theorem ILTD, 1253
inner product
    anti-commutative
        theorem IPAC, 1254
    example CSIP, 1255
    norm
        theorem IPN, 1256
    notation, 1257
    positive
        theorem PIP, 1258
    scalar multiplication
        theorem IPSM, 1259
    vector addition
        theorem IPVA, 1260
integers
    mod p
        definition IMP, 1261
    mod p, field
        theorem FIMP, 1262
    mod 11
        example IM11, 1263
interpolating polynomial
    theorem IP, 1264
invariant subspace
    definition IS, 1265
    eigenspace, 1266
    eigenspaces
        example EIS, 1267
    example TIS, 1268
    Jordan block
        example ISJB, 1269
    kernels of powers
        theorem KPIS, 1270
inverse
    composition of linear transformations
        theorem ICLT, 1271
    example CMI, 1272
    example MI, 1273
    notation, 1274
    of a matrix, 1275
invertible linear transformation
    defined by invertible matrix
        theorem IMILT, 1276
invertible linear transformations
    composition
        theorem CIVLT, 1277
    computing
        example CIVLT, 1278
IP (definition), 1279
IP (notation), 1280
IP (subsection, section O), 1281
IP (theorem), 1282
IPAC (theorem), 1283
IPN (theorem), 1284
IPSM (theorem), 1285
IPVA (theorem), 1286
IS (definition), 1287
IS (example), 1288
IS (section), 1289
IS (subsection, section IS), 1290
ISJB (example), 1291
ISMR4 (example), 1292
ISMR6 (example), 1293
isomorphic
    multiple vector spaces
        example MIVS, 1294
    vector spaces
        example IVSAV, 1295
isomorphic vector spaces
    dimension
        theorem IVSED, 1296
    example TIVS, 1297
ISRN (theorem), 1298
ISSI (example), 1299
ITMT (theorem), 1300
IV (subsection, section IVLT), 1301
IVLT (definition), 1302
IVLT (section), 1303
IVLT (subsection, section IVLT), 1304
IVLT (subsection, section MR), 1305
IVS (definition), 1306
IVSAV (example), 1307
IVSED (theorem), 1308

J (archetype), 1309
JB (definition), 1310
JB (notation), 1311
JB4 (example), 1312
JCF (definition), 1313
JCF (section), 1314
JCF (subsection, section JCF), 1315
JCF10 (example), 1316
JCFLT (theorem), 1317
Jordan block
    definition JB, 1318
    nilpotent
        theorem NJB, 1319
    notation, 1320
    size 4
        example JB4, 1321
Jordan canonical form
    definition JCF, 1322
    size 10
        example JCF10, 1323

K (archetype), 1324
kernel
    injective linear transformation
        theorem KILT, 1325
    isomorphic to null space
        theorem KNSI, 1326
    linear transformation
        example NKAO, 1327
    notation, 1328
    of a linear transformation
        definition KLT, 1329
    pre-image, 1330
    subspace
        theorem KLTS, 1331
    trivial
        example TKAP, 1332
    via matrix representation
        example KVMR, 1333
KILT (theorem), 1334
KLT (definition), 1335
KLT (notation), 1336
KLT (subsection, section ILT), 1337
KLTS (theorem), 1338
KNSI (theorem), 1339
KPI (theorem), 1340
KPIS (theorem), 1341
KPLT (theorem), 1342
KPNLT (example), 1343
KPNLT (theorem), 1344
KVMR (example), 1345

L (archetype), 1346
L (technique, section PT), 1347
LA (subsection, section WILA), 1348
LC (definition), 1349
LC (section), 1350
LC (subsection, section LC), 1351
LC (technique, section PT), 1352
LCCV (definition), 1353
LCM (example), 1354
LDCAA (example), 1355
LDHS (example), 1356
LDP4 (example), 1357
LDRN (example), 1358
LDS (example), 1359
LDS (section), 1360
LDSS (subsection, section LDS), 1361
least squares
    minimizes residuals
        theorem LSMR, 1362
least squares solution
    definition LSS, 1363
left null space
    as row space, 1364
    definition LNS, 1365
    example LNS, 1366
    notation, 1367
    subspace
        theorem LNSMS, 1368
lemma
    technique LC, 1369
LI (definition), 1370
LI (section), 1371
LI (subsection, section LISS), 1372
LIC (example), 1373
LICAB (example), 1374
LICV (definition), 1375
LIHS (example), 1376
LIM32 (example), 1377
linear combination
    system of equations
        example ABLC, 1378
    definition LC, 1379
    definition LCCV, 1380
    example TLC, 1381
    linear transformation, 1382
    matrices
        example LCM, 1383
    system of equations
        example AALC, 1384
linear combinations
    solutions to linear systems
        theorem SLSLC, 1385
linear dependence
    more vectors than size
        theorem MVSLD, 1386
linear independence
    definition LI, 1387
    definition LICV, 1388
    homogeneous systems
        theorem LIVHS, 1389
    injective linear transformation
        theorem ILTLI, 1390
    matrices
        example LIM32, 1391
    orthogonal, 1392
    r and n
        theorem LIVRN, 1393
linear solve
    mathematica, 1394
    sage, 1395
linear system
    consistent
        theorem RCLS, 1396
    matrix representation
        definition MRLS, 1397
        notation, 1398
linear systems
    notation
        example MNSLE, 1399
        example NSLE, 1400
linear transformation
    polynomials to polynomials
        example LTPP, 1401
    addition
        definition LTA, 1402
        theorem MLTLT, 1403
        theorem SLTLT, 1404
    as matrix multiplication
        example ALTMM, 1405
    basis of range
        example BRLT, 1406
    checking
        example ALT, 1407
    composition
        definition LTC, 1408
        theorem CLTLT, 1409
    defined by a matrix
        example LTM, 1410
    defined on a basis
        example LTDB1, 1411
        example LTDB2, 1412
        example LTDB3, 1413
        theorem LTDB, 1414
    definition LT, 1415
    identity
        definition IDLT, 1416
    injection
        definition ILT, 1417
    inverse
        theorem ILTLT, 1418
    inverse of inverse
        theorem IILT, 1419
    invertible
        definition IVLT, 1420
        example AIVLT, 1421
    invertible, injective and surjective
        theorem ILTIS, 1422
    Jordan canonical form
        theorem JCFLT, 1423
    kernels of powers
        theorem KPLT, 1424
    linear combination
        theorem LTLC, 1425
    matrix of, 1426
        example MFLT, 1427
        example MOLT, 1428
    not
        example NLT, 1429
    not invertible
        example ANILT, 1430
    notation, 1431
    polynomials to matrices
        example LTPM, 1432
    rank plus nullity
        theorem RPNDD, 1433
    restriction
        definition LTR, 1434
        notation, 1435
    scalar multiple
        example SMLT, 1436
    scalar multiplication
        definition LTSM, 1437
    spanning range
        theorem SSRLT, 1438
    sum
        example STLT, 1439
    surjection
        definition SLT, 1440
    vector space of, 1441
    zero vector
        theorem LTTZZ, 1442
linear transformation inverse
    via matrix representation
        example ILTVR, 1443
linear transformation restriction
    on generalized eigenspace
        example LTRGE, 1444
linear transformations
    compositions
        example CTLT, 1445
    from matrices
        theorem MBLT, 1446
linearly dependent
    r < n
        example LDRN, 1447
    via homogeneous system
        example LDHS, 1448
linearly dependent columns
    Archetype A
        example LDCAA, 1449
linearly dependent set
    example LDS, 1450
    linear combinations within
        theorem DLDS, 1451
    polynomials
        example LDP4, 1452
linearly independent
    crazy vector space
        example LIC, 1453
    extending sets
        theorem ELIS, 1454
    polynomials
        example LIP4, 1455
    via homogeneous system
        example LIHS, 1456
linearly independent columns
    Archetype B
        example LICAB, 1457
linearly independent set
    example LIS, 1458
    example LLDS, 1459
LINM (subsection, section LI), 1460
LINSB (example), 1461
LIP4 (example), 1462
LIS (example), 1463
LISS (section), 1464
LISV (subsection, section LI), 1465
LIVHS (theorem), 1466
LIVRN (theorem), 1467
LLDS (example), 1468
LNS (definition), 1469
LNS (example), 1470
LNS (notation), 1471
LNS (subsection, section FS), 1472
LNSMS (theorem), 1473
lower triangular matrix
    definition LTM, 1474
LS.MMA (computation, section MMA), 1475
LS.SAGE (computation, section SAGE), 1476
LSMR (theorem), 1477
LSS (definition), 1478
LT (acronyms, section IVLT), 1479
LT (chapter), 1480
LT (definition), 1481
LT (notation), 1482
LT (section), 1483
LT (subsection, section LT), 1484
LTA (definition), 1485
LTC (definition), 1486
LTC (subsection, section LT), 1487
LTDB (theorem), 1488
LTDB1 (example), 1489
LTDB2 (example), 1490
LTDB3 (example), 1491
LTLC (subsection, section LT), 1492
LTLC (theorem), 1493
LTM (definition), 1494
LTM (example), 1495
LTPM (example), 1496
LTPP (example), 1497
LTR (definition), 1498
LTR (notation), 1499
LTRGE (example), 1500
LTSM (definition), 1501
LTTZZ (theorem), 1502

M (acronyms, section FS), 1503
M (archetype), 1504
M (chapter), 1505
M (definition), 1506
M (notation), 1507
MA (definition), 1508
MA (example), 1509
MA (notation), 1510
MACN (Property), 1511
MAF (Property), 1512
MAP (subsection, section SVD), 1513
mathematica
    gram-schmidt (computation), 1514
    linear solve (computation), 1515
    matrix entry (computation), 1516
    matrix inverse (computation), 1517
    matrix multiplication (computation), 1518
    null space (computation), 1519
    row reduce (computation), 1520
    transpose of a matrix (computation), 1521
    vector form of solutions (computation), 1522
    vector linear combinations (computation), 1523
mathematical language
    technique L, 1524
matrix
    addition
        definition MA, 1525
        notation, 1526
    augmented
        definition AM, 1527
    column space
        definition CSM, 1528
    complex conjugate
        example CCM, 1529
    definition M, 1530
    equality
        definition ME, 1531
        notation, 1532
    example AM, 1533
    identity
        definition IM, 1534
    inverse
        definition MI, 1535
    nonsingular
        definition NM, 1536
    notation, 1537
    of a linear transformation
        theorem MLTCV, 1538
    product
        example PTM, 1539
        example PTMEE, 1540
    product with vector
        definition MVP, 1541
    rectangular, 1542
    row space
        definition RSM, 1543
    scalar multiplication
        definition MSM, 1544
        notation, 1545
    singular, 1546
    square
        definition SQM, 1547
    submatrices
        example SS, 1548
    submatrix
        definition SM, 1549
    symmetric
        definition SYM, 1550
    transpose
        definition TM, 1551
    unitary
        definition UM, 1552
    unitary is invertible
        theorem UMI, 1553
    zero
        definition ZM, 1554
matrix addition
    example MA, 1555
matrix components
    notation, 1556
matrix entry
    mathematica, 1557
    sage, 1558
    ti83, 1559
    ti86, 1560
matrix inverse
    Archetype B, 1561
    computation
        theorem CINM, 1562
    mathematica, 1563
    nonsingular matrix
        theorem NI, 1564
    of a matrix inverse
        theorem MIMI, 1565
    one-sided
        theorem OSIS, 1566
    product
        theorem SS, 1567
    sage, 1568
    scalar multiple
        theorem MISM, 1569
    size 2 matrices
        theorem TTMI, 1570
    transpose
        theorem MIT, 1571
    uniqueness
        theorem MIU, 1572
matrix multiplication
    adjoints
        theorem MMAD, 1573
    associativity
        theorem MMA, 1574
    complex conjugation
        theorem MMCC, 1575
    definition MM, 1576
    distributivity
        theorem MMDAA, 1577
    entry-by-entry
        theorem EMP, 1578
    identity matrix
        theorem MMIM, 1579
    inner product
        theorem MMIP, 1580
    mathematica, 1581
    noncommutative
        example MMNC, 1582
    scalar matrix multiplication
        theorem MMSMM, 1583
    systems of linear equations
        theorem SLEMM, 1584
    transposes
        theorem MMT, 1585
    zero matrix
        theorem MMZM, 1586
matrix product
    as composition of linear transformations
        example MPMR, 1587
matrix representation
    basis of eigenvectors
        example MRBE, 1588
    composition of linear transformations
        theorem MRCLT, 1589
    definition MR, 1590
    invertible
        theorem IMR, 1591
    multiple of a linear transformation
        theorem MRMLT, 1592
    notation, 1593
    restriction to generalized eigenspace
        theorem MRRGE, 1594
    sum of linear transformations
        theorem MRSLT, 1595
    theorem FTMR, 1596
    upper triangular
        theorem UTMR, 1597
matrix representations
    converting with change-of-basis
        example MRCM, 1598
    example OLTTR, 1599
matrix scalar multiplication
    example MSM, 1600
matrix vector space
    dimension
        theorem DM, 1601
matrix-adjoint product
    eigenvalues, eigenvectors
        theorem EEMAP, 1602
matrix-vector product
    example MTV, 1603
    notation, 1604
MBC (example), 1605
MBLT (theorem), 1606
MC (notation), 1607
MCC (subsection, section MO), 1608
MCCN (Property), 1609
MCF (Property), 1610
MCN (definition), 1611
MCN (subsection, section CNO), 1612
MCSM (example), 1613
MCT (theorem), 1614
MD (chapter), 1615
ME (definition), 1616
ME (notation), 1617
ME (subsection, section PEE), 1618
ME (technique, section PT), 1619
ME (theorem), 1620
ME.MMA (computation, section MMA), 1621
ME.SAGE (computation, section SAGE), 1622
ME.TI83 (computation, section TI83), 1623
ME.TI86 (computation, section TI86), 1624
MEASM (subsection, section MO), 1625
MFLT (example), 1626
MI (definition), 1627
MI (example), 1628
MI (notation), 1629
MI.MMA (computation, section MMA), 1630
MI.SAGE (computation, section SAGE), 1631
MICN (Property), 1632
MIF (Property), 1633
MIMI (theorem), 1634
MINM (section), 1635
MISLE (section), 1636
MISM (theorem), 1637
MIT (theorem), 1638
MIU (theorem), 1639
MIVS (example), 1640
MLT (subsection, section LT), 1641
MLTCV (theorem), 1642
MLTLT (theorem), 1643
MM (definition), 1644
MM (section), 1645
MM (subsection, section MM), 1646
MM.MMA (computation, section MMA), 1647
MMA (section), 1648
MMA (theorem), 1649
MMAD (theorem), 1650
MMCC (theorem), 1651
MMDAA (theorem), 1652
MMEE (subsection, section MM), 1653
MMIM (theorem), 1654
MMIP (theorem), 1655
MMNC (example), 1656
MMSMM (theorem), 1657
MMT (theorem), 1658
MMZM (theorem), 1659
MNEM (theorem), 1660
MNSLE (example), 1661
MO (section), 1662
MOLT (example), 1663
more variables than equations
    example OSGMD, 1664
    theorem CMVEI, 1665
MPMR (example), 1666
MR (definition), 1667
MR (notation), 1668
MR (section), 1669
MRBE (example), 1670
MRCB (theorem), 1671
MRCLT (diagram), 1672
MRCLT (theorem), 1673
MRCM (example), 1674
MRLS (definition), 1675
MRLS (notation), 1676
MRMLT (theorem), 1677
MRRGE (theorem), 1678
MRS (subsection, section CB), 1679
MRSLT (theorem), 1680
MSCN (example), 1681
MSM (definition), 1682
MSM (example), 1683
MSM (notation), 1684
MTV (example), 1685
multiplicative associativity
    complex numbers
        Property MACN, 1686
multiplicative closure
    complex numbers
        Property MCCN, 1687
    field
        Property MCF, 1688
multiplicative commutativity
    complex numbers
        Property CMCN, 1689
multiplicative inverse
    complex numbers
        Property MICN, 1690
MVNSE (subsection, section RREF), 1691
MVP (definition), 1692
MVP (notation), 1693
MVP (subsection, section MM), 1694
MVSLD (theorem), 1695
MWIAA (example), 1696

N (archetype), 1697
N (subsection, section O), 1698
N (technique, section PT), 1699
NDMS4 (example), 1700
negation of statements
    technique N, 1701
NEM (theorem), 1702
NI (theorem), 1703
NIAO (example), 1704
NIAQ (example), 1705
NIAQR (example), 1706
NIDAU (example), 1707
nilpotent
    linear transformation
        definition NLT, 1708
NILT (diagram), 1709
NJB (theorem), 1710
NJB5 (example), 1711
NKAO (example), 1712
NLT (definition), 1713
NLT (example), 1714
NLT (section), 1715
NLT (subsection, section NLT), 1716
NLTFO (subsection, section LT), 1717
NM (definition), 1718
NM (example), 1719
NM (section), 1720
NM (subsection, section NM), 1721
NM (subsection, section OD), 1722
NM62 (example), 1723
NM64 (example), 1724
NM83 (example), 1725
NME1 (theorem), 1726
NME2 (theorem), 1727
NME3 (theorem), 1728
NME4 (theorem), 1729
NME5 (theorem), 1730
NME6 (theorem), 1731
NME7 (theorem), 1732
NME8 (theorem), 1733
NME9 (theorem), 1734
NMI (subsection, section MINM), 1735
NMLIC (theorem), 1736
NMPEM (theorem), 1737
NMRRI (theorem), 1738
NMTNS (theorem), 1739
NMUS (theorem), 1740
NOILT (theorem), 1741
NOLT (definition), 1742
NOLT (notation), 1743
NOM (definition), 1744
NOM (notation), 1745
nonsingular
    columns as basis
        theorem CNMB, 1746
nonsingular matrices
    linearly independent columns
        theorem NMLIC, 1747
nonsingular matrix
    Archetype B
        example NM, 1748
    column space, 1749
    elementary matrices
        theorem NMPEM, 1750
    equivalences
        theorem NME1, 1751
        theorem NME2, 1752
        theorem NME3, 1753
        theorem NME4, 1754
        theorem NME5, 1755
        theorem NME6, 1756
        theorem NME7, 1757
        theorem NME8, 1758
        theorem NME9, 1759
    matrix inverse, 1760
    null space
        example NSNM, 1761
    nullity, 1762
    product of nonsingular matrices
        theorem NPNT, 1763
    rank
        theorem RNNM, 1764
    row-reduced
        theorem NMRRI, 1765
    trivial null space
        theorem NMTNS, 1766
    unique solutions
        theorem NMUS, 1767
nonsingular matrix, row-reduced
    example NSR, 1768
norm
    example CNSV, 1769
    inner product, 1770
    notation, 1771
normal matrix
    definition NRML, 1772
    example ANM, 1773
    orthonormal basis, 1774
notation
    A, 1775
    AM, 1776
    AME, 1777
    C, 1778
    CCCV, 1779
    CCM, 1780
    CCN, 1781
    CNA, 1782
    CNE, 1783
    CNM, 1784
    CSM, 1785
    CV, 1786
    CVA, 1787
    CVC, 1788
    CVE, 1789
    CVSM, 1790
    D, 1791
    DM, 1792
    DS, 1793
    ELEM, 1794
    ES, 1795
    GES, 1796
    GME, 1797
    HI, 1798
    HID, 1799
    HP, 1800
    IE, 1801
    IM, 1802
    IP, 1803
    JB, 1804
    KLT, 1805
    LNS, 1806
    LT, 1807
    LTR, 1808
    M, 1809
    MA, 1810
    MC, 1811
    ME, 1812
    MI, 1813
    MR, 1814
    MRLS, 1815
    MSM, 1816
    MVP, 1817
    NOLT, 1818
    NOM, 1819
    NSM, 1820
    NV, 1821
    RLT, 1822
    RO, 1823
    ROLT, 1824
    ROM, 1825
    RREFA, 1826
    RSM, 1827
    SC, 1828
    SE, 1829
    SETM, 1830
    SI, 1831
    SM, 1832
    SRM, 1833
    SSET, 1834
    SSV, 1835
    SU, 1836
    SUV, 1837
    T, 1838
    TM, 1839
    VR, 1840
    VSCV, 1841
    VSM, 1842
    ZCV, 1843
    ZM, 1844
notation for a linear system
    example NSE, 1845
NPNT (theorem), 1846
NRFO (subsection, section MR), 1847
NRML (definition), 1848
NRREF (example), 1849
NS.MMA (computation, section MMA), 1850
NSAO (example), 1851
NSAQ (example), 1852
NSAQR (example), 1853
NSC2A (example), 1854
NSC2S (example), 1855
NSC2Z (example), 1856
NSDAT (example), 1857
NSDS (example), 1858
NSE (example), 1859
NSEAI (example), 1860
NSLE (example), 1861
NSLIL (example), 1862
NSM (definition), 1863
NSM (notation), 1864
NSM (subsection, section HSE), 1865
NSMS (theorem), 1866
NSNM (example), 1867
NSNM (subsection, section NM), 1868
NSR (example), 1869
NSS (example), 1870
NSSLI (subsection, section LI), 1871
Null space
    as a span
        example NSDS, 1872
null space
    Archetype I
        example NSEAI, 1873
    basis
        theorem BNS, 1874
    computation
        example CNS1, 1875
        example CNS2, 1876
    isomorphic to kernel, 1877
    linearly independent basis
        example LINSB, 1878
    mathematica, 1879
    matrix
        definition NSM, 1880
    nonsingular matrix, 1881
    notation, 1882
    singular matrix, 1883
    spanning set
        example SSNS, 1884
        theorem SSNS, 1885
    subspace
        theorem NSMS, 1886
null space span, linearly independent
    Archetype L
        example NSLIL, 1887
nullity
    computing, 1888
    injective linear transformation
        theorem NOILT, 1889
    linear transformation
        definition NOLT, 1890
    matrix, 1891
        definition NOM, 1892
    notation, 1893, 1894
    square matrix, 1895
NV (definition), 1896
NV (notation), 1897
NVM (theorem), 1898

O (archetype), 1899
O (Property), 1900
O (section), 1901
OBC (subsection, section B), 1902
OBNM (theorem), 1903
OBUTR (theorem), 1904
OC (Property), 1905
OCN (Property), 1906
OD (section), 1907
OD (subsection, section OD), 1908
OD (theorem), 1909
OF (Property), 1910
OLTTR (example), 1911
OM (Property), 1912
one
    column vectors
        Property OC, 1913
    complex numbers
        Property OCN, 1914
    field
        Property OF, 1915
    matrices
        Property OM, 1916
    vectors
        Property O, 1917
ONFV (example), 1918
ONS (definition), 1919
ONTV (example), 1920
orthogonal
    linear independence
        theorem OSLI, 1921
    set
        example AOS, 1922
    set of vectors
        definition OSV, 1923
    vector pairs
        definition OV, 1924
orthogonal vectors
    example TOV, 1925
orthonormal
    definition ONS, 1926
    matrix columns
        example OSMC, 1927
orthonormal basis
    normal matrix
        theorem OBNM, 1928
orthonormal diagonalization
    theorem OD, 1929
orthonormal set
    four vectors
        example ONFV, 1930
    three vectors
        example ONTV, 1931
OSGMD (example), 1932
OSIS (theorem), 1933
OSLI (theorem), 1934
OSMC (example), 1935
OSV (definition), 1936
OV (definition), 1937
OV (subsection, section O), 1938

P (appendix), 1939
P (archetype), 1940
P (technique, section PT), 1941
particular solutions
    example PSHS, 1942
PCNA (theorem), 1943
PCVS (example), 1944
PD (section), 1945
PDM (section), 1946
PDM (theorem), 1947
PEE (section), 1948
PEEF (theorem), 1949
PI (definition), 1950
PI (subsection, section LT), 1951
PI (technique, section PT), 1952
PIP (theorem), 1953
PM (example), 1954
PM (subsection, section EE), 1955
PMI (subsection, section MISLE), 1956
PMM (subsection, section MM), 1957
PMR (subsection, section MR), 1958
PNLT (subsection, section NLT), 1959
POD (section), 1960
polar decomposition
    theorem PDM, 1961
polynomial
    of a matrix
        example PM, 1962
polynomial vector space
    dimension
        theorem DP, 1963
positive semi-definite
    creating
        theorem CPSM, 1964
positive semi-definite matrix
    definition PSM, 1965
    eigenvalues
        theorem EPSM, 1966
practice
    technique P, 1967
pre-image
    definition PI, 1968
    kernel
        theorem KPI, 1969
pre-images
    example SPIAS, 1970
principal axis theorem, 1971
product of triangular matrices
    theorem PTMT, 1972
Property
    AA, 1973
    AAC, 1974
    AACN, 1975
    AAF, 1976
    AAM, 1977
    AC, 1978
    ACC, 1979
    ACCN, 1980
    ACF, 1981
    ACM, 1982
    AI, 1983
    AIC, 1984
    AICN, 1985
    AIF, 1986
    AIM, 1987
    C, 1988
    CACN, 1989
    CAF, 1990
    CC, 1991
    CM, 1992
    CMCN, 1993
    CMF, 1994
    DCN, 1995
    DF, 1996
    DMAM, 1997
    DSA, 1998
    DSAC, 1999
    DSAM, 2000
    DVA, 2001
    DVAC, 2002
    MACN, 2003
    MAF, 2004
    MCCN, 2005
    MCF, 2006
    MICN, 2007
    MIF, 2008
    O, 2009
    OC, 2010
    OCN, 2011
    OF, 2012
    OM, 2013
    SC, 2014
    SCC, 2015
    SCM, 2016
    SMA, 2017
    SMAC, 2018
    SMAM, 2019
    Z, 2020
    ZC, 2021
    ZCN, 2022
    ZF, 2023
    ZM, 2024
PSHS (example), 2025
PSHS (subsection, section LC), 2026
PSM (definition), 2027
PSM (section), 2028
PSM (subsection, section PSM), 2029
PSM (subsection, section SD), 2030
PSMSR (theorem), 2031
PSPHS (theorem), 2032
PSS (subsection, section SSLE), 2033
PSSD (theorem), 2034
PSSLS (theorem), 2035
PT (section), 2036
PTFP (example), 2037
PTM (example), 2038
PTMEE (example), 2039
PTMT (theorem), 2040

Q (archetype), 2041

R (acronyms, section JCF), 2042
R (archetype), 2043
R (chapter), 2044
R.SAGE (computation, section SAGE), 2045
range
    full
        example FRAN, 2046
    isomorphic to column space
        theorem RCSI, 2047
    linear transformation
        example RAO, 2048
    notation, 2049
    of a linear transformation
        definition RLT, 2050
    pre-image
        theorem RPI, 2051
    subspace
        theorem RLTS, 2052
    surjective linear transformation
        theorem RSLT, 2053
    via matrix representation
        example RVMR, 2054
rank
    computing
        theorem CRN, 2055
    linear transformation
        definition ROLT, 2056
    matrix
        definition ROM, 2057
        example RNM, 2058
    notation, 2059, 2060
    of transpose
        example RRTI, 2061
    square matrix
        example RNSM, 2062
    surjective linear transformation
        theorem ROSLT, 2063
    transpose
        theorem RMRT, 2064
rank one decomposition
    size 2
        example ROD2, 2065
    size 4
        example ROD4, 2066
    theorem ROD, 2067
rank+nullity
    theorem RPNC, 2068
RAO (example), 2069
RCLS (theorem), 2070
RCSI (theorem), 2071
RD (subsection, section VS), 2072
RDS (theorem), 2073
READ (subsection, section B), 2074
READ (subsection, section CB), 2075
READ (subsection, section CRS), 2076
READ (subsection, section D), 2077
READ (subsection, section DM), 2078
READ (subsection, section EE), 2079
READ (subsection, section FS), 2080
READ (subsection, section HSE), 2081
READ (subsection, section ILT), 2082
READ (subsection, section IVLT), 2083
READ (subsection, section LC), 2084
READ (subsection, section LDS), 2085
READ (subsection, section LI), 2086
READ (subsection, section LISS), 2087
READ (subsection, section LT), 2088
READ (subsection, section MINM), 2089
READ (subsection, section MISLE), 2090
READ (subsection, section MM), 2091
READ (subsection, section MO), 2092
READ (subsection, section MR), 2093
READ (subsection, section NM), 2094
READ (subsection, section O), 2095
READ (subsection, section PD), 2096
READ (subsection, section PDM), 2097
READ (subsection, section PEE), 2098
READ (subsection, section RREF), 2099
READ (subsection, section S), 2100
READ (subsection, section SD), 2101
READ (subsection, section SLT), 2102
READ (subsection, section SS), 2103
READ (subsection, section SSLE), 2104
READ (subsection, section TSS), 2105
READ (subsection, section VO), 2106
READ (subsection, section VR), 2107
READ (subsection, section VS), 2108
READ (subsection, section WILA), 2109
reduced row-echelon form
    analysis
        notation, 2110
    definition RREF, 2111
    example NRREF, 2112
    example RREF, 2113
    extended
        definition EEF, 2114
    notation
        example RREFN, 2115
    unique
        theorem RREFU, 2116
reducing a span
    example RSC5, 2117
relation of linear dependence
    definition RLD, 2118
    definition RLDCV, 2119
REM (definition), 2120
REMEF (theorem), 2121
REMES (theorem), 2122
REMRS (theorem), 2123
RES (example), 2124
RGEN (theorem), 2125
rings
    sage, 2126
RLD (definition), 2127
RLDCV (definition), 2128
RLT (definition), 2129
RLT (notation), 2130
RLT (subsection, section IS), 2131
RLT (subsection, section SLT), 2132
RLTS (theorem), 2133
RMRT (theorem), 2134
RNLT (subsection, section IVLT), 2135
RNM (example), 2136
RNM (subsection, section D), 2137
RNNM (subsection, section D), 2138
RNNM (theorem), 2139
RNSM (example), 2140
RO (definition), 2141
RO (notation), 2142
RO (subsection, section RREF), 2143
ROD (section), 2144
ROD (theorem), 2145
ROD2 (example), 2146
ROD4 (example), 2147
ROLT (definition), 2148
ROLT (notation), 2149
ROM (definition), 2150
ROM (notation), 2151
ROSLT (theorem), 2152
row operations
    definition RO, 2153
    elementary matrices, 2154, 2155
    notation, 2156
row reduce
    mathematica, 2157
    sage, 2158
    ti83, 2159
    ti86, 2160
row space
    Archetype I
        example RSAI, 2161
    as column space, 2162
    basis
        example RSB, 2163
        theorem BRS, 2164
    matrix, 2165
    notation, 2166
    row-equivalent matrices
        theorem REMRS, 2167
    subspace
        theorem RSMS, 2168
row-equivalent matrices
    definition REM, 2169
    example TREM, 2170
    row space, 2171
    row spaces
        example RSREM, 2172
    theorem REMES, 2173
row-reduce
    the verb
        definition RR, 2174
row-reduced matrices
    theorem REMEF, 2175
RPI (theorem), 2176
RPNC (theorem), 2177
RPNDD (theorem), 2178
RR (definition), 2179
RR.MMA (computation, section MMA), 2180
RR.SAGE (computation, section SAGE), 2181
RR.TI83 (computation, section TI83), 2182
RR.TI86 (computation, section TI86), 2183
RREF (definition), 2184
RREF (example), 2185
RREF (section), 2186
RREF (subsection, section RREF), 2187
RREFA (notation), 2188
RREFN (example), 2189
RREFU (theorem), 2190
RRTI (example), 2191
RS (example), 2192
RSAI (example), 2193
RSB (example), 2194
RSC4 (example), 2195
RSC5 (example), 2196
RSLT (theorem), 2197
RSM (definition), 2198
RSM (notation), 2199
RSM (subsection, section CRS), 2200
RSMS (theorem), 2201
RSNS (example), 2202
RSREM (example), 2203
RT (subsection, section PD), 2204
RVMR (example), 2205

S (archetype), 2206
S (definition), 2207
S (example), 2208
S (section), 2209
SAA (example), 2210
SAB (example), 2211
SABMI (example), 2212
SAE (example), 2213
sage
    eigenspaces (computation), 2214
    linear solve (computation), 2215
    matrix entry (computation), 2216
    matrix inverse (computation), 2217
    rings (computation), 2218
    row reduce (computation), 2219
    transpose of a matrix (computation), 2220
    vector linear combinations (computation), 2221
SAGE (section), 2222
SAN (example), 2223
SAR (example), 2224
SAS (section), 2225
SAV (example), 2226
SC (definition), 2227
SC (example), 2228
SC (notation), 2229
SC (Property), 2230
SC (subsection, section S), 2231
SC (subsection, section SET), 2232
SC3 (example), 2233
SCAA (example), 2234
SCAB (example), 2235
SCAD (example), 2236
scalar closure
    column vectors
        Property SCC, 2237
    matrices
        Property SCM, 2238
    vectors
        Property SC, 2239
scalar multiple
    matrix inverse, 2240
scalar multiplication
    zero scalar
        theorem ZSSM, 2241
    zero vector
        theorem ZVSM, 2242
    zero vector result
        theorem SMEZV, 2243
scalar multiplication associativity
    column vectors
        Property SMAC, 2244
    matrices
        Property SMAM, 2245
    vectors
        Property SMA, 2246
SCB (theorem), 2247
SCC (Property), 2248
SCM (Property), 2249
SD (section), 2250
SDS (example), 2251
SE (definition), 2252
SE (notation), 2253
secret sharing
    6 ways
        example SS6W, 2254
SEE (example), 2255
SEEF (example), 2256
SER (theorem), 2257
set
    cardinality
        definition C, 2258
        example CS, 2259
        notation, 2260
    complement
        definition SC, 2261
        example SC, 2262
        notation, 2263
    definition SET, 2264
    empty
        definition ES, 2265
    equality
        definition SE, 2266
        notation, 2267
    intersection
        definition SI, 2268
        example SI, 2269
        notation, 2270
    membership
        example SETM, 2271
        notation, 2272
    size, 2273
    subset, 2274
    union
        definition SU, 2275
        example SU, 2276
        notation, 2277
SET (definition), 2278
SET (section), 2279
SETM (example), 2280
SETM (notation), 2281
shoes, 2282
SHS (subsection, section HSE), 2283
SI (definition), 2284
SI (example), 2285
SI (notation), 2286
SI (subsection, section IVLT), 2287
SIM (definition), 2288
similar matrices
    equal eigenvalues
        example EENS, 2289
    eual eigenvalues
        theorem SMEE, 2290
    example SMS3, 2291
    example SMS5, 2292
similarity
    definition SIM, 2293
    equivalence relation
        theorem SER, 2294
singular matrix
    Archetype A
        example S, 2295
    null space
        example NSS, 2296
singular matrix, row-reduced
    example SRR, 2297
singular value decomposition
    theorem SVD, 2298
singular values
    definition SV, 2299
SLE (acronyms, section NM), 2300
SLE (chapter), 2301
SLE (definition), 2302
SLE (subsection, section SSLE), 2303
SLELT (subsection, section IVLT), 2304
SLEMM (theorem), 2305
SLSLC (theorem), 2306
SLT (definition), 2307
SLT (section), 2308
SLTB (theorem), 2309
SLTD (subsection, section SLT), 2310
SLTD (theorem), 2311
SLTLT (theorem), 2312
SM (definition), 2313
SM (notation), 2314
SM (subsection, section SD), 2315
SM2Z7 (example), 2316
SM32 (example), 2317
SMA (Property), 2318
SMAC (Property), 2319
SMAM (Property), 2320
SMEE (theorem), 2321
SMEZV (theorem), 2322
SMLT (example), 2323
SMS (theorem), 2324
SMS3 (example), 2325
SMS5 (example), 2326
SMZD (theorem), 2327
SMZE (theorem), 2328
SNCM (theorem), 2329
SO (subsection, section SET), 2330
socks, 2331
SOL (subsection, section B), 2332
SOL (subsection, section CB), 2333
SOL (subsection, section CRS), 2334
SOL (subsection, section D), 2335
SOL (subsection, section DM), 2336
SOL (subsection, section EE), 2337
SOL (subsection, section F), 2338
SOL (subsection, section FS), 2339
SOL (subsection, section HSE), 2340
SOL (subsection, section ILT), 2341
SOL (subsection, section IVLT), 2342
SOL (subsection, section LC), 2343
SOL (subsection, section LDS), 2344
SOL (subsection, section LI), 2345
SOL (subsection, section LISS), 2346
SOL (subsection, section LT), 2347
SOL (subsection, section MINM), 2348
SOL (subsection, section MISLE), 2349
SOL (subsection, section MM), 2350
SOL (subsection, section MO), 2351
SOL (subsection, section MR), 2352
SOL (subsection, section NM), 2353
SOL (subsection, section O), 2354
SOL (subsection, section PD), 2355
SOL (subsection, section PDM), 2356
SOL (subsection, section PEE), 2357
SOL (subsection, section RREF), 2358
SOL (subsection, section S), 2359
SOL (subsection, section SD), 2360
SOL (subsection, section SLT), 2361
SOL (subsection, section SS), 2362
SOL (subsection, section SSLE), 2363
SOL (subsection, section T), 2364
SOL (subsection, section TSS), 2365
SOL (subsection, section VO), 2366
SOL (subsection, section VR), 2367
SOL (subsection, section VS), 2368
SOL (subsection, section WILA), 2369
solution set
    Archetype A
        example SAA, 2370
    archetype E
        example SAE, 2371
    theorem PSPHS, 2372
solution set of a linear system
    definition SSSLE, 2373
solution sets
    possibilities
        theorem PSSLS, 2374
solution to a linear system
    definition SSLE, 2375
solution vector
    definition SOLV, 2376
SOLV (definition), 2377
solving homogeneous system
    Archetype A
        example HISAA, 2378
    Archetype B
        example HUSAB, 2379
    Archetype D
        example HISAD, 2380
solving nonlinear equations
    example STNE, 2381
SP4 (example), 2382
span
    basic
        example ABS, 2383
    basis
        theorem BS, 2384
    definition SS, 2385
    definition SSCV, 2386
    improved
        example IAS, 2387
    notation, 2388
    reducing
        example RSC4, 2389
    reduction
        example RS, 2390
    removing vectors
        example COV, 2391
    reworking elements
        example RES, 2392
    set of polynomials
        example SSP, 2393
    subspace
        theorem SSS, 2394
span of columns
    Archetype A
        example SCAA, 2395
    Archetype B
        example SCAB, 2396
    Archetype D
        example SCAD, 2397
spanning set
    crazy vector space
        example SSC, 2398
    definition TSVS, 2399
    matrices
        example SSM22, 2400
    more vectors
        theorem SSLD, 2401
    polynomials
        example SSP4, 2402
SPIAS (example), 2403
SQM (definition), 2404
square root
    eigenvalues, eigenspaces
        theorem EESR, 2405
    matrix
        definition SRM, 2406
        notation, 2407
    positive semi-definite matrix
        theorem PSMSR, 2408
    unique
        theorem USR, 2409
SR (section), 2410
SRM (definition), 2411
SRM (notation), 2412
SRM (subsection, section SR), 2413
SRR (example), 2414
SS (definition), 2415
SS (example), 2416
SS (section), 2417
SS (subsection, section LISS), 2418
SS (theorem), 2419
SS6W (example), 2420
SSC (example), 2421
SSCV (definition), 2422
SSET (definition), 2423
SSET (example), 2424
SSET (notation), 2425
SSLD (theorem), 2426
SSLE (definition), 2427
SSLE (section), 2428
SSM22 (example), 2429
SSNS (example), 2430
SSNS (subsection, section SS), 2431
SSNS (theorem), 2432
SSP (example), 2433
SSP4 (example), 2434
SSRLT (theorem), 2435
SSS (theorem), 2436
SSSLE (definition), 2437
SSSLT (subsection, section SLT), 2438
SSV (notation), 2439
SSV (subsection, section SS), 2440
standard unit vector
    notation, 2441
starting proofs
    technique GS, 2442
STLT (example), 2443
STNE (example), 2444
SU (definition), 2445
SU (example), 2446
SU (notation), 2447
submatrix
    notation, 2448
subset
    definition SSET, 2449
    notation, 2450
subspace
    as null space
        example RSNS, 2451
    characterized
        example ASC, 2452
    definition S, 2453
    in P4
        example SP4, 2454
    not, additive closure
        example NSC2A, 2455
    not, scalar closure
        example NSC2S, 2456
    not, zero vector
        example NSC2Z, 2457
    testing
        theorem TSS, 2458
    trivial
        definition TS, 2459
    verification
        example SC3, 2460
        example SM32, 2461
subspaces
    equal dimension
        theorem EDYES, 2462
surjective
    Archetype N
        example SAN, 2463
    example SAR, 2464
    not
        example NSAQ, 2465
        example NSAQR, 2466
    not, Archetype O
        example NSAO, 2467
    not, by dimension
        example NSDAT, 2468
    polynomials to matrices
        example SAV, 2469
surjective linear transformation
    bases
        theorem SLTB, 2470
surjective linear transformations
    dimension
        theorem SLTD, 2471
SUV (definition), 2472
SUV (notation), 2473
SUVB (theorem), 2474
SUVOS (example), 2475
SV (definition), 2476
SVD (section), 2477
SVD (subsection, section SVD), 2478
SVD (theorem), 2479
SVP4 (example), 2480
SYM (definition), 2481
SYM (example), 2482
symmetric matrices
    theorem SMS, 2483
symmetric matrix
    example SYM, 2484
system of equations
    vector equality
        example VESE, 2485
system of linear equations
    definition SLE, 2486

T (archetype), 2487
T (definition), 2488
T (notation), 2489
T (part), 2490
T (section), 2491
T (technique, section PT), 2492
TCSD (example), 2493
TD (section), 2494
TD (subsection, section TD), 2495
TD (theorem), 2496
TD4 (example), 2497
TDEE (theorem), 2498
TDEE6 (example), 2499
TDSSE (example), 2500
TDSSE (subsection, section TD), 2501
technique
    C, 2502
    CD, 2503
    CP, 2504
    CV, 2505
    D, 2506
    DC, 2507
    E, 2508
    GS, 2509
    I, 2510
    L, 2511
    LC, 2512
    ME, 2513
    N, 2514
    P, 2515
    PI, 2516
    T, 2517
    U, 2518
theorem
    AA, 2519
    AIP, 2520
    AISM, 2521
    AIU, 2522
    AMA, 2523
    AMSM, 2524
    BCS, 2525
    BIS, 2526
    BNS, 2527
    BRS, 2528
    BS, 2529
    CB, 2530
    CCM, 2531
    CCRA, 2532
    CCRM, 2533
    CCT, 2534
    CFDVS, 2535
    CFNLT, 2536
    CHT, 2537
    CILTI, 2538
    CINM, 2539
    CIVLT, 2540
    CLI, 2541
    CLTLT, 2542
    CMVEI, 2543
    CNMB, 2544
    COB, 2545
    CPSM, 2546
    CRMA, 2547
    CRMSM, 2548
    CRN, 2549
    CRSM, 2550
    CRVA, 2551
    CSCS, 2552
    CSLTS, 2553
    CSMS, 2554
    CSNM, 2555
    CSRN, 2556
    CSRST, 2557
    CSS, 2558
    CUMOS, 2559
    DC, 2560
    DCM, 2561
    DCP, 2562
    DEC, 2563
    DED, 2564
    DEM, 2565
    DEMMM, 2566
    DER, 2567
    DERC, 2568
    DFS, 2569
    DGES, 2570
    DIM, 2571
    DLDS, 2572
    DM, 2573
    DMFE, 2574
    DMHP, 2575
    DMMP, 2576
    DMST, 2577
    DNLT, 2578
    DP, 2579
    DRCM, 2580
    DRCMA, 2581
    DRCS, 2582
    DRMM, 2583
    DSD, 2584
    DSFB, 2585
    DSFOS, 2586
    DSLI, 2587
    DSZI, 2588
    DSZV, 2589
    DT, 2590
    DVM, 2591
    DZRC, 2592
    EDELI, 2593
    EDYES, 2594
    EEMAP, 2595
    EER, 2596
    EESR, 2597
    EIM, 2598
    EIS, 2599
    ELIS, 2600
    EMDRO, 2601
    EMHE, 2602
    EMMVP, 2603
    EMN, 2604
    EMNS, 2605
    EMP, 2606
    EMRCP, 2607
    EMS, 2608
    ENLT, 2609
    EOMP, 2610
    EOPSS, 2611
    EPM, 2612
    EPSM, 2613
    ERMCP, 2614
    ESMM, 2615
    ETM, 2616
    FIMP, 2617
    FS, 2618
    FTMR, 2619
    FVCS, 2620
    G, 2621
    GEK, 2622
    GESD, 2623
    GESIS, 2624
    GSP, 2625
    HMIP, 2626
    HMOE, 2627
    HMRE, 2628
    HMVEI, 2629
    HPC, 2630
    HPDAA, 2631
    HPHI, 2632
    HPHID, 2633
    HPSMM, 2634
    HSC, 2635
    ICBM, 2636
    ICLT, 2637
    IFDVS, 2638
    IILT, 2639
    ILTB, 2640
    ILTD, 2641
    ILTIS, 2642
    ILTLI, 2643
    ILTLT, 2644
    IMILT, 2645
    IMR, 2646
    IP, 2647
    IPAC, 2648
    IPN, 2649
    IPSM, 2650
    IPVA, 2651
    ISRN, 2652
    ITMT, 2653
    IVSED, 2654
    JCFLT, 2655
    KILT, 2656
    KLTS, 2657
    KNSI, 2658
    KPI, 2659
    KPIS, 2660
    KPLT, 2661
    KPNLT, 2662
    LIVHS, 2663
    LIVRN, 2664
    LNSMS, 2665
    LSMR, 2666
    LTDB, 2667
    LTLC, 2668
    LTTZZ, 2669
    MBLT, 2670
    MCT, 2671
    ME, 2672
    MIMI, 2673
    MISM, 2674
    MIT, 2675
    MIU, 2676
    MLTCV, 2677
    MLTLT, 2678
    MMA, 2679
    MMAD, 2680
    MMCC, 2681
    MMDAA, 2682
    MMIM, 2683
    MMIP, 2684
    MMSMM, 2685
    MMT, 2686
    MMZM, 2687
    MNEM, 2688
    MRCB, 2689
    MRCLT, 2690
    MRMLT, 2691
    MRRGE, 2692
    MRSLT, 2693
    MVSLD, 2694
    NEM, 2695
    NI, 2696
    NJB, 2697
    NME1, 2698
    NME2, 2699
    NME3, 2700
    NME4, 2701
    NME5, 2702
    NME6, 2703
    NME7, 2704
    NME8, 2705
    NME9, 2706
    NMLIC, 2707
    NMPEM, 2708
    NMRRI, 2709
    NMTNS, 2710
    NMUS, 2711
    NOILT, 2712
    NPNT, 2713
    NSMS, 2714
    NVM, 2715
    OBNM, 2716
    OBUTR, 2717
    OD, 2718
    OSIS, 2719
    OSLI, 2720
    PCNA, 2721
    PDM, 2722
    PEEF, 2723
    PIP, 2724
    PSMSR, 2725
    PSPHS, 2726
    PSSD, 2727
    PSSLS, 2728
    PTMT, 2729
    RCLS, 2730
    RCSI, 2731
    RDS, 2732
    REMEF, 2733
    REMES, 2734
    REMRS, 2735
    RGEN, 2736
    RLTS, 2737
    RMRT, 2738
    RNNM, 2739
    ROD, 2740
    ROSLT, 2741
    RPI, 2742
    RPNC, 2743
    RPNDD, 2744
    RREFU, 2745
    RSLT, 2746
    RSMS, 2747
    SCB, 2748
    SER, 2749
    SLEMM, 2750
    SLSLC, 2751
    SLTB, 2752
    SLTD, 2753
    SLTLT, 2754
    SMEE, 2755
    SMEZV, 2756
    SMS, 2757
    SMZD, 2758
    SMZE, 2759
    SNCM, 2760
    SS, 2761
    SSLD, 2762
    SSNS, 2763
    SSRLT, 2764
    SSS, 2765
    SUVB, 2766
    SVD, 2767
    TD, 2768
    TDEE, 2769
    technique T, 2770
    TIST, 2771
    TL, 2772
    TMA, 2773
    TMSM, 2774
    TSE, 2775
    TSRM, 2776
    TSS, 2777
    TT, 2778
    TTMI, 2779
    UMCOB, 2780
    UMI, 2781
    UMPIP, 2782
    USR, 2783
    UTMR, 2784
    VFSLS, 2785
    VRI, 2786
    VRILT, 2787
    VRLT, 2788
    VRRB, 2789
    VRS, 2790
    VSLT, 2791
    VSPCV, 2792
    VSPM, 2793
    ZSSM, 2794
    ZVSM, 2795
    ZVU, 2796
ti83
    matrix entry (computation), 2797
    row reduce (computation), 2798
    vector linear combinations (computation), 2799
TI83 (section), 2800
ti86
    matrix entry (computation), 2801
    row reduce (computation), 2802
    transpose of a matrix (computation), 2803
    vector linear combinations (computation), 2804
TI86 (section), 2805
TIS (example), 2806
TIST (theorem), 2807
TIVS (example), 2808
TKAP (example), 2809
TL (theorem), 2810
TLC (example), 2811
TM (definition), 2812
TM (example), 2813
TM (notation), 2814
TM (subsection, section OD), 2815
TM.MMA (computation, section MMA), 2816
TM.SAGE (computation, section SAGE), 2817
TM.TI86 (computation, section TI86), 2818
TMA (theorem), 2819
TMP (example), 2820
TMSM (theorem), 2821
TOV (example), 2822
trace
    definition T, 2823
    linearity
        theorem TL, 2824
    matrix multiplication
        theorem TSRM, 2825
    notation, 2826
    similarity
        theorem TIST, 2827
    sum of eigenvalues
        theorem TSE, 2828
trail mix
    example TMP, 2829
transpose
    matrix scalar multiplication
        theorem TMSM, 2830
    example TM, 2831
    matrix addition
        theorem TMA, 2832
    matrix inverse, 2833, 2834
    notation, 2835
    scalar multiplication, 2836
transpose of a matrix
    mathematica, 2837
    sage, 2838
    ti86, 2839
transpose of a transpose
    theorem TT, 2840
TREM (example), 2841
triangular decomposition
    entry by entry, size 6
        example TDEE6, 2842
    entry by entry
        theorem TDEE, 2843
    size 4
        example TD4, 2844
    solving systems of equations
        example TDSSE, 2845
    theorem TD, 2846
triangular matrix
    inverse
        theorem ITMT, 2847
trivial solution
    system of equations
        definition TSHSE, 2848
TS (definition), 2849
TS (subsection, section S), 2850
TSE (theorem), 2851
TSHSE (definition), 2852
TSM (subsection, section MO), 2853
TSRM (theorem), 2854
TSS (section), 2855
TSS (subsection, section S), 2856
TSS (theorem), 2857
TSVS (definition), 2858
TT (theorem), 2859
TTMI (theorem), 2860
TTS (example), 2861
typical systems, 2 × 2
    example TTS, 2862

U (archetype), 2863
U (technique, section PT), 2864
UM (definition), 2865
UM (subsection, section MINM), 2866
UM3 (example), 2867
UMCOB (theorem), 2868
UMI (theorem), 2869
UMPIP (theorem), 2870
unique solution, 3 × 3
    example US, 2871
    example USR, 2872
uniqueness
    technique U, 2873
unit vectors
    basis
        theorem SUVB, 2874
    definition SUV, 2875
    orthogonal
        example SUVOS, 2876
unitary
    permutation matrix
        example UPM, 2877
    size 3
        example UM3, 2878
unitary matrices
    columns
        theorem CUMOS, 2879
unitary matrix
    inner product
        theorem UMPIP, 2880
UPM (example), 2881
upper triangular matrix
    definition UTM, 2882
US (example), 2883
USR (example), 2884
USR (theorem), 2885
UTM (definition), 2886
UTMR (subsection, section OD), 2887
UTMR (theorem), 2888

V (acronyms, section O), 2889
V (archetype), 2890
V (chapter), 2891
VA (example), 2892
Vandermonde matrix
    definition VM, 2893
vandermonde matrix
    determinant
        theorem DVM, 2894
    nonsingular
        theorem NVM, 2895
    size 4
        example VM4, 2896
VEASM (subsection, section VO), 2897
vector
    addition
        definition CVA, 2898
    column
        definition CV, 2899
    equality
        definition CVE, 2900
        notation, 2901
    inner product
        definition IP, 2902
    norm
        definition NV, 2903
    notation, 2904
    of constants
        definition VOC, 2905
    product with matrix, 2906, 2907
    scalar multiplication
        definition CVSM, 2908
vector addition
    example VA, 2909
vector component
    notation, 2910
vector form of solutions
    Archetype D
        example VFSAD, 2911
    Archetype I
        example VFSAI, 2912
    Archetype L
        example VFSAL, 2913
    example VFS, 2914
    mathematica, 2915
    theorem VFSLS, 2916
vector linear combinations
    mathematica, 2917
    sage, 2918
    ti83, 2919
    ti86, 2920
vector representation
    example AVR, 2921
    example VRC4, 2922
    injective
        theorem VRI, 2923
    invertible
        theorem VRILT, 2924
    linear transformation
        definition VR, 2925
        notation, 2926
        theorem VRLT, 2927
    surjective
        theorem VRS, 2928
    theorem VRRB, 2929
vector representations
    polynomials
        example VRP2, 2930
vector scalar multiplication
    example CVSM, 2931
vector space
    characterization
        theorem CFDVS, 2932
    column vectors
        definition VSCV, 2933
    definition VS, 2934
    infinite dimension
        example VSPUD, 2935
    linear transformations
        theorem VSLT, 2936
    over integers mod 5
        example VSIM5, 2937
vector space of column vectors
    notation, 2938
vector space of functions
    example VSF, 2939
vector space of infinite sequences
    example VSIS, 2940
vector space of matrices
    definition VSM, 2941
    example VSM, 2942
    notation, 2943
vector space of polynomials
    example VSP, 2944
vector space properties
    column vectors
        theorem VSPCV, 2945
    matrices
        theorem VSPM, 2946
vector space, crazy
    example CVS, 2947
vector space, singleton
    example VSS, 2948
vector spaces
    isomorphic
        definition IVS, 2949
        theorem IFDVS, 2950
VESE (example), 2951
VFS (example), 2952
VFSAD (example), 2953
VFSAI (example), 2954
VFSAL (example), 2955
VFSLS (theorem), 2956
VFSS (subsection, section LC), 2957
VFSS.MMA (computation, section MMA), 2958
VLC.MMA (computation, section MMA), 2959
VLC.SAGE (computation, section SAGE), 2960
VLC.TI83 (computation, section TI83), 2961
VLC.TI86 (computation, section TI86), 2962
VM (definition), 2963
VM (section), 2964
VM4 (example), 2965
VO (section), 2966
VOC (definition), 2967
VR (definition), 2968
VR (notation), 2969
VR (section), 2970
VR (subsection, section LISS), 2971
VRC4 (example), 2972
VRI (theorem), 2973
VRILT (theorem), 2974
VRLT (theorem), 2975
VRP2 (example), 2976
VRRB (theorem), 2977
VRS (theorem), 2978
VS (acronyms, section PD), 2979
VS (chapter), 2980
VS (definition), 2981
VS (section), 2982
VS (subsection, section VS), 2983
VSCV (definition), 2984
VSCV (example), 2985
VSCV (notation), 2986
VSF (example), 2987
VSIM5 (example), 2988
VSIS (example), 2989
VSLT (theorem), 2990
VSM (definition), 2991
VSM (example), 2992
VSM (notation), 2993
VSP (example), 2994
VSP (subsection, section MO), 2995
VSP (subsection, section VO), 2996
VSP (subsection, section VS), 2997
VSPCV (theorem), 2998
VSPM (theorem), 2999
VSPUD (example), 3000
VSS (example), 3001

W (archetype), 3002
WILA (section), 3003

X (archetype), 3004

Z (Property), 3005
ZC (Property), 3006
ZCN (Property), 3007
ZCV (definition), 3008
ZCV (notation), 3009
zero
    complex numbers
        Property ZCN, 3010
    field
        Property ZF, 3011
zero column vector
    definition ZCV, 3012
    notation, 3013
zero matrix
    notation, 3014
zero vector
    column vectors
        Property ZC, 3015
    matrices
        Property ZM, 3016
    unique
        theorem ZVU, 3017
    vectors
        Property Z, 3018
ZF (Property), 3019
ZM (definition), 3020
ZM (notation), 3021
ZM (Property), 3022
ZNDAB (example), 3023
ZSSM (theorem), 3024
ZVSM (theorem), 3025
ZVU (theorem), 3026